Twelve identical resistors, each of resistance 12 ohm, are soldered…
Question
Twelve identical resistors, each of resistance 12 ohm, are soldered along the twelve edges of a cube. A battery is connected between two vertices that are diametrically opposite (the two ends of a space/body diagonal of the cube). Determine the equivalent resistance, in ohm, between these two vertices.
Step-by-step solution
Inject current I at vertex A and remove it at the opposite vertex G.
By symmetry the current I splits equally into the 3 edges leaving A (each I/3).
At the next layer (3 vertices), each of those currents splits, and by symmetry each of the 6 middle edges carries I/6.
Finally 3 edges feed into G each carrying I/3.
The potential drop from A to G along any path A->mid->mid->G is V = (I/3)R + (I/6)R + (I/3)R = R(I/3 + I/6 + I/3) = (5/6)IR.
Final answer10
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